On the probabilities of moderate deviations for sums of independent random variables

Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables and $\sigma>0$. Put
$$ F_n(x)=\mathbf P\biggl\{\sum_{i=1}^nX_i<x\biggr\},\qquad\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^x e^{-t^2/2}\,dt. $$
Necessary and sufficient conditions are found for the validity of the relation
$$ 1-F_n(x\sigma\sqrt n)=(1-\Phi(x))(1+o(1)),\qquad 0\le x\le c\sqrt{\log n},\qquad n\to\infty. $$